Abstract
For f(x) , a separable polynomial of degree d over a discretely valued field K , we describe how the cluster picture of f(x) over K , in other words, the set of tuples {(ord(xi - xj), i, j):1 ≤ i < j ≤ d} , where x1, …, xd are the roots of f(x) , can be recovered without knowing the roots of f(x) over K¯ . We construct an explicit list of polynomials gd(1), …, gd(td) ∈ ℤ[A0, …, Ad-1] such that the valuations ord(gd(i)(a0, …, ad-1)) for i = 1, …, td uniquely determine this set of distances for the polynomial f(x) = cf(xd + ad-1xd-1 + ⋯ + a0) , and we describe the process by which they do so. We use this to deduce that if C:y2 = f(x) is a hyperelliptic curve over a local field K . This list of valuations of polynomials in the coefficients of f(x) uniquely determines the dual graph of the special fibre of the minimal strict normal crossings model of C/Kunr , the inertia action on the Tate module and the conductor exponent. This provides a hyperelliptic curves analogue to a corollary of Tate's algorithm, that in residue characteristic p ≥ 5 , the dual graph of special fibre of the minimal regular model of an elliptic curve E/Kunr is uniquely determined by the valuation of jE and ΔE .